keyword:dumb - cp's OEIS frontend

A057946 Bad hexadecimal notation for n: write n in hexadecimal notation, replacing digit ten with "10", digit eleven with "11", ... and digit fifteen with "15".

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 110, 111, 112, 113, 114, 115, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 210, 211, 212, 213, 214, 215, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 310, 311, 312, 313, 314, 315, 40, 41

Offset: 0

Henry Bottomley, Oct 13 2000

			a(15)=15 since it is represented by the digit "15" base 16. a(21)=15 since it is represented by the digit "1" followed by the digit "5" base 16.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A057947 for ambiguous numbers, A055645 for a better representation of hexadecimal.

a:= proc(n) local i, m, r ; m:=n; r:=NULL;
      if n=0 then return 0 fi;
      for i from 0 while m>0 do
        r:= irem(m, 16, 'm'), r
      od; parse(cat(r))
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Jun 23 2014

f[n_] := FromDigits@ Flatten@ IntegerDigits@ IntegerDigits[n, 16]; Array[f, 66, 0] (* Robert G. Wilson v, Jun 23 2014 *)

A107625 Every digit of prime and its index contains a loop (only digits 0,4,6,8,9 in prime and its index).

Original entry on oeis.org

409, 4409, 4999, 6869, 44699, 44909, 46499, 48409, 64849, 66889, 68449, 68909, 86969, 89689, 98869, 98899, 480499, 488689, 490499, 609809, 806609, 806999, 866969, 868669, 868849, 869489, 869849, 869899, 869909, 880949, 6440809, 6440999, 6488969, 6489449

Offset: 1

Zak Seidov, May 18 2005

Corresponding indices in A107624. Cf. A001744 Every digit contains a loop.

Harvey P. Dale, Table of n, a(n) for n = 1..376

Cf. A001744, A107624.

Do[id=Union[IntegerDigits[p=Prime[n]], IntegerDigits[n]];If[Count[id, 1]+Count[id, 2]+Count[id, 3]+Count[id, 5]+Count[id, 7]==0, Print[p]], {n, 10000}]
Module[{c={0,4,6,8,9}},Select[Prime/@(Rest[FromDigits/@Tuples[c,6]]), SubsetQ[ c,IntegerDigits[#]]&]] (* Harvey P. Dale, Sep 03 2015 *)

is_a001744(n) = #setintersect(vecsort(digits(n), , 8), [1, 2, 3, 5, 7])==0
my(i=1); forprime(p=1, 65e5, if(is_a001744(p) && is_a001744(i), print1(p, ", ")); i++) \\ Felix Fröhlich, Sep 09 2019

More terms from Harvey P. Dale, Sep 03 2015

A138563 Beastly fax numbers: numbers containing the fax number of the Beast (667, one more than its regular number) in their decimal expansion.

Original entry on oeis.org

667, 1667, 2667, 3667, 4667, 5667, 6667, 6670, 6671, 6672, 6673, 6674, 6675, 6676, 6677, 6678, 6679, 7667, 8667, 9667, 10667, 11667, 12667, 13667, 14667, 15667, 16667, 16670, 16671, 16672, 16673, 16674, 16675, 16676, 16677, 16678

Offset: 1

N. J. A. Sloane, May 13 2007

The sum of the reciprocals of numbers not in this sequence is convergent. - Adam P. Goucher, Apr 27 2014

Jens Kruse Andersen, Table of n, a(n) for n = 1..1000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Margaret Wertheim and Neil Sloane, The Fax Numbers of the Beast, and Other Mathematical Sports: An Interview with Neil Sloane, Cabinet, Spring 2015.

Cf. A051003.

Select[Range[20000], StringContainsQ[ToString[#], "667"] &] (* Amiram Eldar, Jun 28 2024 *)

a(n) ~ n. - Charles R Greathouse IV, Oct 25 2014

Sum_{k>=1, k is not a term} 1/k = 2301.846622336249707557560554200194249235044868457872023381489896767824372028... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Jun 28 2024

A215009 Numbers which are "easy" to key on a computer numpad.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 20, 21, 23, 25, 32, 36, 41, 45, 47, 52, 54, 56, 58, 63, 65, 69, 74, 78, 85, 87, 89, 96, 98, 101, 120, 121, 123, 125, 141, 145, 147, 202, 210, 212, 214, 232, 236, 252, 254, 256, 258, 320, 321, 323, 325, 363, 365, 369, 410

Offset: 1

Arkadiusz Wesolowski, Jul 31 2012

On a computer numpad, a number is "easy" to key in if each adjacent pair of digits in the number are adjacent - either horizontally or vertically.
Here are two ways to type these numbers. Example for 25:
1. Press the numpad "2" key. Then let go and press "5".
2. Press the "2" and slide your finger on the numeric keypad up to the "5".
Method 2 shows that the sequence contains only numbers in which every pair of adjacent digits are distinct.
Pressing a numeric key followed by "0" and pressing a numeric key is equivalent to selecting "101" or "202".

			25 is a term because the 2 and 5 keys are adjacent.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
Wikipedia, Numeric keypad
Index entries for 10-automatic sequences.

Cf. A082390. Subsequence of A043096.

lst = {}; Do[If[StringCount[ToString[n], {"00", "03", "04", "05", "06", "07", "08", "09", "11", "13", "15", "16", "17", "18", "19", "22", "24", "26", "27", "28", "29", "30", "31", "33", "34", "35", "37", "38", "39", "40", "42", "43", "44", "46", "48", "49", "50", "51", "53", "55", "57", "59", "60", "61", "62", "64", "66", "67", "68", "70", "71", "72", "73", "75", "76", "77", "79", "80", "81", "82", "83", "84", "86", "88", "90", "91", "92", "93", "94", "95", "97", "99", "102", "201"}] == 0, AppendTo[lst, n]], {n, 0, 410}]; lst

from itertools import count, islice
m = {'0':'12', '1':'024', '2':'0135', '3':'26', '4':'157', '5':'2468', '6':'359', '7':'48', '8':'579', '9':'68'}
def c(r): return (r=='0' or r[0]!='0') and not ("102" in r or "201" in r)
def agen():
    reach = list("0123456789")
    for d in count(1):
        yield from (int(r) for r in reach if c(r))
        reach = [r + s for r in reach for s in m[r[-1]]]
print(list(islice(agen(), 62))) # Michael S. Branicky, Jul 05 2022

A049004 First letter of English names for months of year, mapping A -> 1, B -> 2 etc.

Original entry on oeis.org

10, 6, 13, 1, 13, 10, 10, 1, 19, 15, 14, 4, 10, 6, 13, 1, 13, 10, 10, 1, 19, 15, 14, 4, 10, 6, 13, 1, 13, 10, 10, 1, 19, 15, 14, 4, 10, 6, 13, 1, 13, 10, 10, 1, 19, 15, 14, 4, 10, 6, 13, 1, 13, 10, 10, 1, 19, 15, 14, 4, 10, 6, 13, 1, 13, 10, 10, 1, 19, 15, 14, 4, 10, 6, 13, 1, 13, 10

Offset: 1

Deepak R. N (deepak_rama(AT)bigfoot.com)

Period 12: repeat [10, 6, 13, 1, 13, 10, 10, 1, 19, 15, 14, 4]. - Joerg Arndt, Aug 15 2014

			a(7)=10 because July begins with the 10th letter.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

LetterNumber[DateValue[#,"MonthNameInitial"]]&/@DateRange[{2024,1,1},{2030,12,31},"Month"] (* or *) PadRight[{},120,{10,6,13,1,13,10,10,1,19,15,14,4}] (* Harvey P. Dale, Sep 10 2024 *)

Corrected a(12n+8): August starts with 'A' and not 'H'. - Mark E. Shoulson, Aug 15 2014

a(20) corrected by Daniel Leary, Jul 26 2016

A090805 A simple recurrence with one error.

Original entry on oeis.org

1, 2, 6, 21, 85, 430, 2586, 18109, 144880, 1303929, 13039300, 143432311, 1721187744, 22375440685, 313256169604, 4698842544075, 75181480705216, 1278085171988689, 23005533095796420, 437105128820131999, 8742102576402640000, 183584154104455440021, 4038851390298019680484

Offset: 0

N. J. A. Sloane, Feb 12 2004

I included this in the OEIS only because was published on a web page. The explanation is my own - perhaps the original proposer had a different explanation.

			1..add.1..multiply.by 1 -> 2
2..add.1..multiply.by 2 -> 6
6......1............. 3 -> 21
21.....1............. 4 -> 88 but here you make a mistake and instead multiply by 4 and add 1, getting 85
85.....1............. 5 -> 430
430....1............. 6 -> 2586
etc

Found on a puzzle page.

Hugo Delestinne, Meerdaelquiz
N. J. A. Sloane and Brady Haran, A Sequence with a Mistake, Numberphile video (2021)

Cf. A033540, A344229.

a:= proc(n) a(n):= n*a(n-1) + `if`(n=4, 1, n) end: a(0):= 1:
seq(a(n), n=0..22);  # Alois P. Heinz, May 14 2021

a={1};Do[n=Length[a];a=Append[a,If[n==4,Last[a]n+1,(Last[a]+1)n]],22];a (* Jake L Lande, Jul 28 2024 *)

a(0) = 1; a(n) = n*(a(n-1) + 1) but make an error if n = 4.

Hans Havermann points out that the first 7 terms could also be produced by the recurrence f[x] = f[x - 1]*(x - 1) + GCD[3*f[x - 1], (x - 1)] with f[1] = 1. (This gives the continuation 1, 2, 6, 21, 85, 430, 2586, 18103, 144825, 1303434, 13034342, ...) But given the nature of the other problems on this quiz, I think my explanation is more likely.

A107624 Numbers n such that every digit of n and n-th prime contains a loop (only digits 0,4,6,8,9 in n and n-th prime).

Original entry on oeis.org

80, 600, 669, 884, 4646, 4666, 4806, 4980, 6480, 6666, 6806, 6849, 8448, 8688, 9489, 9494, 40046, 40664, 40804, 49848, 64444, 64466, 68864, 68994, 69008, 69060, 69084, 69089, 69090, 69899, 440986, 440999, 444049, 444080, 464446, 464496, 464499, 466466, 466844

Offset: 1

Zak Seidov, May 18 2005

Corresponding primes in A107625. Cf. A001744 Every digit contains a loop.

Cf. A001744, A107625.

Do[id=Union[IntegerDigits[Prime[n]], IntegerDigits[n]];If[Count[id, 1]+Count[id, 2]+Count[id, 3]+Count[id, 5]+Count[id, 7]==0, Print[n]], {n, 10000}]

is_a001744(n) = #setintersect(vecsort(digits(n), , 8), [1, 2, 3, 5, 7])==0
is(n) = is_a001744(n) && is_a001744(prime(n)) \\ Felix Fröhlich, Sep 09 2019

More terms from Felix Fröhlich, Sep 09 2019

A139138 Numbers divisible by at least two of their digits.

Original entry on oeis.org

11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 101, 102, 104, 105, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 131, 132, 135, 138, 140, 141, 142, 144, 145, 147, 148, 150, 151, 152, 153, 155, 156, 161, 162

Offset: 1

Jonathan Vos Post, Jun 05 2008

Digits need not be distinct. This may be considered row 2 of an infinite array whose 1st row is A038770. Each such row is a subset of the ones above it.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Alvin Hoover Belt)
Index entries for 10-automatic sequences

Cf. A038770, A187516.

fQ[n_] := Count[Mod[n, Flatten[IntegerDigits[n] /. {0 -> {}}]], 0] > 1; Select[ Range@ 170, fQ] (* Robert G. Wilson v, Jun 23 2014 *)
Select[Range[200],Count[Divisible[#,Select[IntegerDigits[#], #>0&]], True]>1&] (* Harvey P. Dale, Dec 16 2015 *)

from sympy import factorint
def ok(n): return sum(1 for d in map(int, str(n)) if d > 0 and n%d == 0) > 1
print([k for k in range(163) if ok(k)]) # Michael S. Branicky, Nov 12 2021

More terms from Alvin Hoover Belt, Apr 06 2009

Own omission (140) fixed by Alvin Hoover Belt, Apr 18 2009

A229381 The Simpsons's perfect number, Mersenne prime, and narcissistic number.

Original entry on oeis.org

8128, 8191, 8208

Offset: 1

Joe Sondow and Jonathan Sondow, Sep 23 2013

The perfect number 8128, Mersenne prime 8191, and narcissistic number 8208 appeared together on the screen in the "Marge and Homer Turn a Couple Play" episode of Season 17 of The Simpsons.

S. Singh, The Simpsons and Their Mathematical Secrets, Bloomsbury Publishing, London, 2013, pp. 93-96, ISBN 9781408835302 / 9781408843734.

C. Goff, Animating Popular Mathematics: "The Simpsons and Their Mathematical Secrets", AMS Notices, Vol. 62, No. 1 (2015), 40-44.
Simon Singh, The Simpsons' secret formula: it's written by maths geeks, The Guardian, 21 September 2013.

Cf. A000396, A000668, A005188.

a(1) = A000396(4), a(2) = A000668(5), a(3) = A005188(15).

A039790 Prime numbers prefixed with a '1'.

Original entry on oeis.org

12, 13, 15, 17, 111, 113, 117, 119, 123, 129, 131, 137, 141, 143, 147, 153, 159, 161, 167, 171, 173, 179, 183, 189, 197, 1101, 1103, 1107, 1109, 1113, 1127, 1131, 1137, 1139, 1149, 1151, 1157, 1163, 1167, 1173, 1179, 1181, 1191, 1193, 1197, 1199, 1211

Offset: 1

Kevin N. Stone (kevin.stone(AT)brainbashers.com)

Replace every prime by the concatenation of its divisors. [Lekraj Beedassy, May 29 2009]

David F. Marrs, Table of n, a(n) for n = 1..10000
Kevin N. Stone, What number comes next in this sequence, BrainBashers [Archived copy at the Wayback Machine].

Cf. A000040, A037278, A289866 (terms which are prime).

Array[10^Floor[1 + Log10[#]] + # &@ Prime[#] &, 47] (* Michael De Vlieger, Apr 05 2021 *)

a(n) = eval(concat(Str(1), Str(prime(n)))) \\ Felix Fröhlich, Apr 05 2021

a(n) = A037278(A000040(n)). [Lekraj Beedassy, May 29 2009]

cp's OEIS Frontend

A057946 Bad hexadecimal notation for n: write n in hexadecimal notation, replacing digit ten with "10", digit eleven with "11", ... and digit fifteen with "15".

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A107625 Every digit of prime and its index contains a loop (only digits 0,4,6,8,9 in prime and its index).

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A138563 Beastly fax numbers: numbers containing the fax number of the Beast (667, one more than its regular number) in their decimal expansion.

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A215009 Numbers which are "easy" to key on a computer numpad.

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A049004 First letter of English names for months of year, mapping A -> 1, B -> 2 etc.

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A090805 A simple recurrence with one error.

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A107624 Numbers n such that every digit of n and n-th prime contains a loop (only digits 0,4,6,8,9 in n and n-th prime).

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A139138 Numbers divisible by at least two of their digits.

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